Another vote for The Ivy. It’s lovely and classy and the food is always delicious.
This doesn’t tie in to “what to eat pre-theatre,” but if you are in central London and want the best fish and chips in town, try Golden Union, at 38 Poland Street, near Oxford Circus. It’s a small family-run place and their fish and chips are really sublime. You can take away, but there is a cute small space to dine in in the back. The jukebox is free. :-)
For a pub meal that is elevated a step or 3 above regular “pub grub,” you might try The Devonshire, at 17 Denman Street, just off Piccadilly Circus. They pour a perfect pint of Guinness, age their own beef, and get in super high quality seafood. Staff are very friendly. They haven’t been open long and were BUSY busy busy at first: I’d consider making a reservation for dinner. This doesn’t qualify as an “old pub,” then - but it’s very, very nice and the publican is a top bloke.
https://www.devonshiresoho.co.uk/
The Lamb and Flag, at 33 Rose Street, is a very old pub with lots of history. It oozes with “ye olde” charm on the inside. I’ve had many a pint here, but never a meal, so I can’t weigh in on the food.
https://www.lambandflagcoventgarden.co.uk/
Assume you are consulting the RS guide book for sights to see? If you tell us more specific interests, we could probably be of better help.
Oh! Theatre! Catching a play at The Globe is an amazing experience, and the company puts on very high quality productions of the classics. It’s great. I also recommend The Donmar Warehouse in Covent Garden for fabulous new productions in an intimate space. This is more serious theatre, usually, not glitzy West End popular entertainment (she said snobbishly ;-) Although they do mount comedies, too. :-)
https://www.donmarwarehouse.com/
If you are anywhere in central London there will be a LOT of tourists. That is unavoidable. But it is completely possible to avoid tourist traps. Have a great time! :-)